Hydrostatic Balance of an Adiabatic Fluid in a Background Potential

Consider an adiabatic fluid with an adiabatic index
. We want to compute the density structure
in a potential . The gas
pressure is .

First, note that
or

The equation of vertical hydrostatic equilibrium is

If we define the central midplane gas temperature at
to be such that the sound speed is , then . Given the above, we then require

We can arrange this if we define and write

and our density profile becomes

A numerical integral sets by demanding

Hydrostatic disks with varying surface densities

So we can use the above to set the central density if the surface density is not a function of radius. But the
surface density is declining with radius and the potential is varying as the radius increases.

Once we set at , , we are stuck with that equation of state if the disk gas is all
on the same adiabat.

Now, reconsider a different place in the disk. The central
density in the midplane will be lower because of the
surface density, but also lower if the potential has changed radially.

Clearly , so the constant
of integration in determining the density by integrating
the vertical force is not the same as at .

Instead we can write

Where

such that . The density
where the sound speed is is a constant, so the surface density constraint must
be used to set . This has to be done iteratively, but a first reasonable guess is